#6469252

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,23,29,24,30,22,28)(19,31,32,26,21,27,20,25);(9,10)(11,16)(12,14)(13,15)(17,19,24,21)(18,25,30,26)(20,22,32,23)(27,28,31,29);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20,24,32)(18,26,30,25)(19,22,21,23)(27,28,31,29);(10,14)(13,16)(17,21,24,19)(18,27,30,31)(20,22,32,23)(25,28,26,29);(1,4,5,3)(2,6,7,8)(17,22,24,23)(18,28,30,29)(19,32,21,20)(25,31,26,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376448);